I experienced the same problem with a 1 year, 1 month old Cinema Display connected to an early 2013 MacBook Air (13"). Initially I was able to disconnect the MagSafe adapter and fiddle with the connection which would temporarily solve the problem for me. Eventully, the fidlling solution stopped working and I needed a better one (for my ~1 year old $1,000 monitor!).
After reading through this thread, I first tried cleaning the contacts on my MagSafe, Converter and MBA (with a Q tip), then, after that didn't work I headed to an Apple store and pick up a new MagSafe converter (based on suggestions from others here).
I spoke with a Genius who told me he has a few friends who own 2013 MBA's and MBP's with Retina that had faulty motherboards which he said could be the problem here, and he told me to bring in my MBA in to run diagnostics and determine if that's the case with mine, but I couldn't make an appointment for a few days and wanted to see what I could do sooner.
When I told him I read suggestions to get a new MagSafe converter, he agreed it was worth a try and so I bought one and guess what?! It seems to have solved my problem (granted I've only been using it for a day now). That said, if you try a new MagSafe Converter and it doesn't work (and you own a 2013+ MBA or MBP with Retina), bring the laptop to your local apple store so they can run a diagnostic to determine if you have a faulty motherboard, because if you do, apple should replace it free of charge in the first year.
Good luck!